Spectral Compact Training: Pre-Training Large Language Models via Permanent Truncated SVD and Stiefel QR Retraction

The memory wall remains the primary bottleneck for training large language models on consumer hardware. We introduce Spectral Compact Training (SCT), a method that replaces dense weight matrices with permanent truncated SVD factors W = U diag(s) V^T, where the full dense matrix is never materialized during training or inference. Gradients flow through the compact spectral factors via standard backpropagation, and U, V are retracted to the Stiefel manifold via QR decomposition after each optimizer step. SCT achieves up to 199x memory reduction per MLP layer at rank 32, enabling full training steps of 70B-parameter architectures on a Steam Deck handheld (7.2 GB peak memory vs. 1,245 GB for dense FP32 training with Adam). Rank-sweep experiments on SmolLM2-1.7B (ranks 32-256, 2000 steps, NVIDIA A100) show that all tested ranks converge to the same loss floor (~4.2-4.5), identifying the learning rate schedule -- not MLP rank -- as the primary bottleneck. Rank 128 emerges as the efficiency sweet spot at 11.7x MLP compression with the lowest perplexity. GPU memory drops 46% at rank 32 while training throughput doubles.